Suppose that random variables $X_1$ and $X_2$ only take values on the non-negative integers, and they satisfy the monotone likelihood ratio property, i.e. $\forall x>y$, $\frac{f_1(x)}{f_2(x)}\geq \frac{f_1(y)}{f_2(y)}$, where $f_1$ is the pdf of $X_1$ and $f_2$ is the pdf of $X_2$.
Show that $\frac{E[X_1]}{E[\sqrt{X_1}]} > \frac{E[X_2]}{E[\sqrt{X_2}]}$.
Would this work for any ratio of moments where the numerator is a higher moment than the denominator? Thank you for your help.
EDIT: Removed question about whether first-order stochastic dominance is sufficient for this to be true. It is not. Counter-example: consider $f_1(0)=0.1,f_1(1)=0.2,f_1(2)=0.7,f_2(0)=0.2,f_2(1)=0.1,f_2(2)=0.7$.